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One final thing: what mathematical background would one need to prove the result we want to (regarding the orthogonality requirement of an IRF)? Would linear algebra (up to basic knowledge of covectors / vectors and transformation properties of tensors) suffice? Or would it require more advanced stuff like differential geometry and require me to know fancy concepts like Ricci curvature, etc.? I'll accordingly brush up on the needed mathematical prerequisites.Here is an exercise that may help to clarify the above statements of mine.

Consider Rindler coordinates [1]. This is a different coordinate chart on Minkowski spacetime, in which the metric of Minkowski spacetime has a different representation that is not ##\text{diag}(-1,1,1,1)##.

However, this coordinate chart is non-inertial. Try showing this by showing that this chart does not meet either of the two requirements for an IRF: free particles do not have constant coordinate velocity, and light does not always have speed ##c##.

[1] https://en.wikipedia.org/wiki/Rindler_coordinates